What is the integral of (x^nln(x)) ?

The integral of (x^nln(x)) is 1/(n+1)x^{n+1}ln(x)-(x^{n+1})/((n+1)^2)+C

Popular >

integral of (x^nln(x))

\int (x^{n} ln (x)) d x

\frac{1}{n + 1} x^{n + 1} ln (x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}} + C

Solution steps

\int x^{n} ln (x) d x

Apply Integration By Parts

: \frac{1}{n + 1} x^{n + 1} ln (x) - \int \frac{x ^{n}}{n + 1} d x

= \frac{1}{n + 1} x^{n + 1} ln (x) - \int \frac{x ^{n}}{n + 1} d x

\int \frac{x ^{n}}{n + 1} d x = \frac{x ^{n + 1}}{( n + 1 ) ^{2}}

= \frac{1}{n + 1} x^{n + 1} ln (x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}}

Add a constant to the solution

= \frac{1}{n + 1} x^{n + 1} ln (x) - \frac{x ^{n + 1}}{( n + 1 ) ^{2}} + C

\int_{- 2}^{0} 3 w 4 - w^{2} d w

t \to 0 lim (\frac{e ^{t} - 1}{t})

f (x) = (\frac{x}{7} + \frac{7}{x})^{7}

\frac{d ^{4}}{d x ^{4}} (ln (1 - x))

\int_{1}^{2} \frac{5}{8 x ^{6}} d x

What is the integral of (x^nln(x)) ?
The integral of (x^nln(x)) is 1/(n+1)x^{n+1}ln(x)-(x^{n+1})/((n+1)^2)+C